Revista Colombiana de Estadística
Junio 2013, volumen 36, no. 1, pp. 23 a 42
A Multi-Stage Almost Ideal Demand System: The
Case of Beef Demand in Colombia
Sistema casi ideal de demanda multinivel: el caso de la demanda de
carne de res en Colombia
Andrés Ramírez a
Departamento de Economía, Universidad EAFIT, Medellín, Colombia
Abstract
The main objective in this paper is to obtain reliable long-term and shortterm elasticities estimates of the beef demand in Colombia using quarterly
data since 1998 until 2007. However, complexity on the decision process of
consumption should be taken into account, since expenditure on a particular
good is sequential. In the case of beef demand in Colombia, a Multi-Stage
process is proposed based on an Almost Ideal Demand System (AIDS). The
econometric novelty in this paper is to estimate simultaneously all the stages
by the Generalized Method of Moments to obtain a joint covariance matrix
of parameter estimates in order to use the Delta Method for calculating the
standard deviation of the long-term elasticities estimates. Additionally, this
approach allows us to get elasticity estimates in each stage, but also, total
elasticities which incorporate interaction between stages. On the other hand,
the short-term dynamic is handled by a simultaneous estimation of the Error
Correction version of the model; therefore, Monte Carlo simulation exercises
are performed to analyse the impact on beef demand because of shocks at
different levels of the decision making process of consumers. The results indicate that, although the total expenditure elasticity estimate of demand for
beef is 1.78 in the long-term and the expenditure elasticity estimate within
the meat group is 1.07, the total short-term expenditure elasticity is merely
0.03. The smaller short-term reaction of consumers is also evidenced on price
shocks; while the total own price elasticity of beef is -0.24 in the short-term,
the total and within meat group long-term elasticities are −1.95 and −1.17,
respectively.
Key words: Cointegration, Delta method, Demand system, Generalized
method of moments, Monte Carlo Simulation.
a Associate
professor. E-mail: aramir21@eafit.edu.co
23
24
Andrés Ramírez
Resumen
El objetivo más importante de este artículo es obtener estimaciones confiables de las elasticidades de la demanda de carne de res en Colombia para el
largo y corto plazo utilizando información trimestral desde 1998 hasta 2007.
Sin embargo, las decisiones que toman los consumidores se enmarcan en un
ambiente complejo, puesto que el gasto en un bien particular se realiza de
forma secuencial. En el caso particular de la demanda de carne de res en la
economía colombiana, se propone un Sistema Casi Ideal de Demanda Multinivel. La novedad econométrica en este artículo es estimar simulatáneamente
todos los niveles del modelo mediante el Método Generalizado de los Momentos; esto permite obtener una matriz conjunta de covarianzas de todos
los parámetros, y así utilizar el Método Delta para calcular las desviaciones
estándar de las elasticidades estimadas de largo plazo. Adicionalmente, este
enfoque nos permite obtener estimaciones de las elasticidades en cada nivel,
pero también, elasticidades totales que incorporan la interacción entre los
niveles. Por otra parte, la dinámica de corto plazo se estudia a través de la
estimación conjunta de la versión en Corrección de Errores del modelo; de
esta forma, ejercicios de simulación Monte Carlo son reaizados para analizar
el impacto sobre la demanda de carne de res debido a perturbaciones en
diferentes niveles del proceso de toma de decisiones de los consumidores.
Los resultados indican que aunque en el largo plazo la elasticidad estimada
de la demanda de carne de res con respecto al gasto total es 1.78, y la elasticidad estimada de la demanda con respecto al gasto en cárnicos es 1.07,
la elasticidad de la demanda con respecto al gasto total en el corto plazo es
solo 0.03. La reducida reacción en el corto plazo también está presente ante
perturbaciones en el precio; mientras que la elasticidad precio propia total de
la demanda de carne de res es −0.24 en el corto plazo, las elasticidades total
y al interior del grupo de cárnicos para el largo plazo son −1.95 y −1.17,
respectivamente.
Palabras clave: cointegración, método delta, método generalizado de los
momentos, simulación Monte Carlo, sistema de demanda.
1. Introduction
Colombian beef demand is important for a number of reasons. Historically
consumers have generally preferred beef to other types of meat. Beef accounted
for approximately 60% of the total meat budget, compared to only 30% for poultry and 10% for pork. In addition, the beef sector is an important component
of the Colombian economy, accounting for 3.4% of Gross Domestic Product in
2007 and providing 1.4 million jobs (DANE 2007). Moreover, the beef sector is a
significant component of the Colombian exports to Venezuela, one of Colombia’s
most important trading partners. Approximately, 15% of Colombian beef production is exported to Venezuela. Recently, the Venezuelan Government decided to
stop imports from Colombia as a result of political tensions. This trade restriction policy of Venezuela has generated preoccupation among specialists due to its
consequences for the beef sector. Additionally, Colombia is currently negotiating
international trade agreements with the United States and the European Union.
Revista Colombiana de Estadística 36 (2013) 23–42
Multi-Stage AIDS for Beef Demand in Colombia
25
The implication is that the Colombian beef sector would have international competition from countries with high subsidies, as a consequence, given the trading
conditions, the internal beef price would decrease. On the other hand, there is
an asymmetric aspect that is necessary to take into account, the Colombian beef
sector does not have international certification on phytosanitary aspects while the
United States and the European Union accomplish this requirement. This implies
that Colombia cannot export beef while the latter countries can do it. All these
changes would, in turn, affect internal beef demand. So on the whole, understanding beef demand is necessary for the Colombian agricultural policy.
Although the beef sector is important for the Colombian economy, little effort
has been made to estimate demand elasticities and simulate different scenarios
that impact on the sector. Therefore from an economic point of view, the objective of this study is to obtain reliable estimates of Colombian meat demand,
and make some simulation exercises in order to evaluate the impact of different
shocks on beef demand. Given that policy evaluations and simulations require reliable estimates of demand responsiveness to price and expenditure (Wahl, Hayes &
Williams 1991), the methodology used to estimate elasticities is the Almost Ideal
Demand System (AIDS), because
“. . . gives an arbitrary first-order approximation to any demand system; it satisfies the axioms of choice exactly; it aggregates perfectly
over consumers without invoking parallel linear Engel curves; it has
a functional form which is consistent with known household-budget
data; it is simple to estimate, largely avoiding the need for non-linear
estimation; and it can be used to test the restriction of homogeneity
and symmetry through linear restrictions on fixed parameters.”
(Deaton & Muellbauer 1980a, pp 312)
Specifically, we use a Multi-Stage AIDS model due to consumers following
multiple steps when acquiring goods in the market. This approach allows us to
estimate long-term elasticities in each stage, and also, total elasticities which incorporate interaction between levels. Additionally from an econometric perspective,
it is well known that the level of uncertainty associated with elasticities estimates
is very important; therefore, a simultaneous estimation procedure permits us to
estimate a joint covariance matrix which can be used to calculate the standard
deviation of the elasticities through the Delta Method. This is the methodological
novelty of our paper. In particular, we use the Generalized Method of Moments
to estimate the complete system.
Referring to short-term dynamics, we estimate an Error Correction version
of the Multi-Stage Almost Ideal Demand System, and then, we simulate shocks
at different levels of the decision making process of the consumers and measure
their impacts. This strategy allows us to calculate, the short-term impact on beef
demand associated with changes in the consumer’s total expenditure and prices of
beef, poultry and pork.
There is extensive empirical literature on the demand for meat. In most of this
literature, the demand is estimated using the AIDS methodology (Asatryan 2003,
Revista Colombiana de Estadística 36 (2013) 23–42
26
Andrés Ramírez
Clark 2006, Fuller 1997, Galvis 2000, Holt & Goodwin 2009, Sulgham & Zapata
2006). Even though there have been efforts in Colombia to determine beef demand
elasticities (Caraballo 2003, Galvis 2000) most of the literature is focused on North
America and Asia. Due undoubtedly to widely varying economic conditions across
countries, the estimates of the elasticities of demand vary greatly. For example,
the expenditure elasticity of beef consumption varies between 0.23 and 1.68. In
the wealthier countries in the West, it is often below 1.0 (Barreira & Duarte 1997,
Clark 2006, MAFF 2000, Sulgham & Zapata 2006), while in the poorer countries
in the East it is generally above 1.0 (Liu, Parton, Zhou & Cox 2008, Chern,
Ishibashi, Taniguchi & Tokoyama 2003, Ma, Huang, Rozelle & Rae 2003, Rastegari
& Hwang 2007). The own-Marshallian price demand elasticity is between −1.19
and −0.10, usually less than -1 (Fousekis & Revell 2000, Galvis 2000, Golan, Perloff
& Shen 2000). The compensated price elasticities show that changes in price does
not affect the demand for beef as much.
In the specific case of Colombia, Galvis (2000) estimated the elasticities of demand for beef, poultry, and pork using the Seemingly Unrelated Regression (SUR)
technique. He estimated an expenditure elasticity of demand for beef between 0.67
and 0.79, while the Marshallian (own price) elasticity is between −1.19 and −1.41.
The cross-price elasticity of poultry prices on beef demand is between 0.27 and
0.96, and the cross-price elasticity of pork on beef demand is between 1.08 and
1.37. However, Galvis (2000) did not perform unit root tests, so the regressions
might be spurious in the event that the variables are not cointegrated.
The empirical results in this article indicate that the long-term total and within
meat group uncompensated price elasticities are −1.95 and −1.17, respectively.
The total and within group compensated price elasticities are −1.78 and −0.52,
and the total consumer expenditure elasticity of demand is 1.78. The results also
indicate that consumers substitute beef for poultry, but not beef for pork. The
short-term elasticities, calculated through Monte Carlo simulations, are smaller.
They indicate that an increase of 1% in the price of beef decreases its demand by
0.24%, while increasing total expenditure by 1% has no significant impact on the
demand for beef in Colombia.
The paper is organized as follows. Section 2 provides the methodology, Section
3 presents the long-term results, Section 4 presents some Monte Carlo simulation
exercises, and Section 5 concludes.
2. Methodology
The methodology used in this paper is based on a Multi-Stage model which
replicates the decision making process of the consumers when they buy beef
(Gao, Eric, Gail & Cramer 1996, Michalek & Keyzer 1992, Shenggen, Wailes &
Cramer 1995). Necessary and sufficient conditions for estimating a Multi-Stage
budgeting process are that the direct utility function must be additively separable and the specific satisfaction functions in each stage should be homogeneous.
Gorman (1957) provided conditions for this procedure to be optimal subject to
the condition that must have more than two groups in each stage. Blackorby &
Revista Colombiana de Estadística 36 (2013) 23–42
Multi-Stage AIDS for Beef Demand in Colombia
27
Russell (1997), extends Gorman’s classic result to encompass the two-group cases
that he did not take into account. These conditions are very restrictive, and must
be in general considered implausible. However, Edgerton (1997) showed that a
Multi-Stage budgeting process will lead to an approximately correct allocation if
preferences are weakly separable and the group price indices being used do not
vary too greatly with utility level. This means that a change in price of a commodity in one group affects the demand for all commodities in another group in
the same manner. Also that the group price indices do not vary too greatly with
expenditure level.
In particular, we estimate a Multi-stage Ideal Demand System of three levels to
obtain the long-term elasticities in each level, and also, the total elasticities. The
complete system is estimated using the Generalized Method of Moments. Following this strategy, the resulting three problems will be smaller and more tractable
from an empirical point of view than the original problem, because including all
goods prices in each of the equations is often faced with the problem of having too
many variables (Segerson & Mount 1985). The long-term estimation is based on
equation (1).
In order to simulate shocks in the short-term at different levels of the decision
making process of consumers, we estimate the Error Correction version of the
Multi-Stage AIDS model. This strategy allow us to calculate by Monte Carlo
simulations, the short-term impact on beef demand associated with changes in
the consumer’s total expenditure and the prices of beef, poultry and pork. This
estimation is based on equation (11).
This strategy considers the complex decision process through which an individual makes consumption decisions. Specifically, there are three levels: The upper
one determines the aggregate level of food consumption; the middle one, conditioned by the upper one, determines the consumption of meat, and the lower level,
conditioned by the other two, determines the beef, poultry, and pork demand.
In order to handle each stage budgeting process, an Almost Ideal Demand System is introduced (Deaton & Muellbauer 1980a). The mathematical specification
of the AIDS model is the following,
wit = αi +
N
X
γij ln(pjt ) + βi ln(Xt /Pt ) + eit
(1)
j=1
for i = 1, 2, . . . , N , j = 1, 2, . . . , N and t = 1, 2, . . . , T where N is the number of
goods, T is the temporal length, and the share in the total expenditure of the good
i (wit ) is a function of the prices (pjt ), real expenditure(Xt /Pt ) and an error (eit ).
The general price index is usually represented by a nonlinear equation which is, in
most cases, replaced by the Stone price index
ln(PtS ) =
N
X
wit ln(pit )
(2)
i=1
However, the Stone index typically used in estimating Linear AIDS is not invariant
to changes in units of measurement, which may seriously affect the approximation
Revista Colombiana de Estadística 36 (2013) 23–42
28
Andrés Ramírez
properties of the model and can result in biased parameter estimates (Pashardes
1993, Moschini 1995). To overcome this problem other specifications for the price
index can be used, such as the Paasche (3) or Laspeyres (4) index:
ln(PtP ) =
N
X
wit ln(pit /p0i )
(3)
i=1
ln(PtL ) =
N
X
wi0 ln(pit )
(4)
i=1
where the superscript represents a base period.
It is worth noting the constraints (additivity, homogeneity and symmetry) that
are imposed by the microeconomic theory:
N
X
αi = 1,
i=1
N
X
γij = 0,
i=1
N
X
N
X
βi = 0
(5)
i=1
γij = 0
(6)
j=1
γij = γji
(7)
From the above specification the following long-term elasticities in each level
can be calculated:
ηit = 1 + βi /wit
(8)
M
ijt = −IA + γij /wit − βi (wjt /wit )
(9)
H
ijt = −IA + γij /wit + wjt
(10)
where IA = 1 if i = j.
H
Where ηit , M
ijt and ijt are expenditure, Marshallian (uncompensated) and
Hicksian (compensated) elasticities, respectively.
It is required to investigate the time series properties of the data used in order to
specify the most appropriate dynamic form of the model and to find out if the longterm demand relationships provided by equation (1) are economically meaningful
or they are merely spurious. If all variables in equation (1) are cointegrated, the
Error Correction Linear AIDS is given by the following form:
∆wit =
N
X
j=1
δij ∆wjt−1 +
N
X
γij ∆ln(pjt ) + βi ∆ln(Xt /Pt ) + λêi,t−1 + µit ,
(11)
j=1
for i = 1, 2, . . . , N , j = 1, 2, . . . , N y t = 1, 2, . . . , T , where ∆ refers to the difference
operator, êi,t−1 represents the estimated residuals from the cointegrated equation
(1), −1 < λ < 0 is the velocity of convergence, and µit is the error term. IntertemPN
poral consistency requires that
& Blundell 1983) and
i=1 δij = 0 (Anderson
PN
identification of the lagged budget shares requires j=1 δij = 0 (Edgerton 1997).
Revista Colombiana de Estadística 36 (2013) 23–42
29
Multi-Stage AIDS for Beef Demand in Colombia
Once the cointegrated equations are estimated, we can calculate the long-term
total demand elasticities. Edgerton (1997) provide expressions to get elasticities
associated with the lower level and we adapt these equations as follows:
(T )
ηit = ηit × ηM eat,t × ηF ood,t
(12)
M (T )
H
M
= H
ijt + wjt × ηit × M eat,t + wjt × wM eat,t × ηit × ηM eat,t × F ood,t (13)
H(T )
H
H
= H
ijt + wjt × ηit × M eat,t + wjt × wM eat,t × ηit × ηM eat,t × F ood,t (14)
ijt
ijt
where superscript, i, j = beef, pork, poultry.
(T )
The total expenditure elasticity of beef demand, ηit , is a product of the
expenditure elasticity of food, the food expenditure elasticity of meat and the
M (T )
H(T )
meat expenditure elasticity of beef. The total price elasticities, ijt and ijt ,
are the result of a direct effect within the meat group, but also of the reallocation
effects of meat within food, and food within total consumption. Finally, we obtain
standard deviations for the total elasticities with the Delta Method where this
method establishes that given Z = (Z1 , Z2 , . . . , Zk ), a random vector with mean
θ = (θ1 , θ2 , . . . , θk ), if g(Z) is a differentiable function, we can approximate its
variance by
V arθ g(Z) ≈
k
X
X
(gi0 (θ))2 V arθ (Zi ) + 2
gi0 (θ)gj0 (θ)Covθ (Zi , Zj )
i=1
i>j
where gi0 (θ) =
∂
∂zi g(z)|z1 =θ1 ,z2 =θ2 ,...,zk =θk .
(T )
= ηi = ηi × ηM eat × ηF ood ,
Let g(Z)
the total expenditure elasticity in the
lower stage. We approximate its variance by
V
(T )
arθ ηi
2
1
≈
(ηM eat ηF ood ) V ar(βi )
wi
2
1
(ηi ηF ood ) V ar(βM eat )
+
wM eat
2
1
(ηi ηM eat ) V ar(βF ood )
+
wF ood
1
2
+ 2
(ηi ηM eat )(ηF ood ) Cov(βi , βM eat )
wi wM eat
1
2
+ 2
(ηi ηF ood )(ηM eat ) Cov(βi , βF ood )
wi wF ood
1
+ 2
(ηM eat ηF ood )(ηi )2 Cov(βM eat , βF ood )
wM eat wF ood
where θ = (βi , βM eat , βF ood ), and i, j = beef, pork, poultry.
Revista Colombiana de Estadística 36 (2013) 23–42
30
Andrés Ramírez
It must be observed that we need the covariance between the expenditure
parameters at different stages. Therefore, we have to estimate the three levels
simultaneously.
M (T )
Now let g(Z) = ijt
M
F ood,t i.e.,
M (T )
ij
H
= H
ijt +wjt ×ηit ×M eat,t +wjt ×wM eat,t ×ηit ×ηM eat,t ×
=
(−IA + γij /wi + wj )
+
wj (1 + βi /wi )(−1 + γM eat /wM eat + wM eat )
+
wj wM eat (1 + βi /wi )(1 + βM eat /wM eat )(−1 + γF ood /wF ood − βF ood )
We can approximate the variance of the Marshallian total price demand elasticity by
2
1
M (T )
V ar(γij )
V arθ ij
≈
wi
2
wj wM eat
wj H
M
+
ηM eat F ood V ar(βi )
+
wi M eat
wi
2
wj
ηi V ar(γM eat )
+
wM eat
2
+ wj ηi M
V ar(βM eat )
F ood
2
wj wM eat
ηi ηF ood V ar(γF ood )
+
wF ood
+
+
+
+
+
+
+
+
+
2
(−wj wM eat ηi ηF ood ) V ar(βF ood )
wj H
wj wM eat
M
2
+
η
M eat F ood Cov(γij , βi )
(wi )2 M eat
(wi )2
wj
2
ηi Cov(γij , γM eat )
wi wM eat
wj M
ηi F ood Cov(γij , βM eat )
2
wi
wj wM eat
ηi ηF ood Cov(γij , γF ood )
2
wi wF ood
−wj wM eat
2
ηi ηF ood Cov(γij , βF ood )
wi
wj H
wj wM eat
wj
2
M eat +
ηM eat M
η
i Cov(βi , γM eat )
F ood
wi
wi
wM eat
wj H
wj wM eat
2
M eat +
ηM eat M
wj ηi M
F ood
F ood Cov(βi , βM eat )
wi
wi
wj H
wj wM eat
wj wM eat
2
M eat +
ηM eat M
η
η
Cov(βi , γF ood )
i
F
ood
F ood
wi
wi
wF ood
Revista Colombiana de Estadística 36 (2013) 23–42
Multi-Stage AIDS for Beef Demand in Colombia
31
wj H
wj wM eat
M
2
+
ηM eat F ood (−wj wM eat ηi ηF ood ) Cov(βi , βF ood )
wi M eat
wi
wj
2
ηi wj ηi M
F ood Cov(γM eat , βM eat )
wM eat
wj
wj wM eat
2
ηi
ηi ηF ood Cov(γM eat , γF ood )
wM eat
wF ood
wj
2
ηi (−wj wM eat ηi ηF ood ) Cov(γM eat , βF ood )
wM eat
wj wM eat
M
2 wj ηi F ood
ηi ηF ood Cov(βM eat , γF ood )
wF ood
2 wj ηi M
F ood (−wj wM eat ηi ηF ood ) Cov(βM eat , βF ood )
wj wM eat
2
ηi ηF ood (−wj wM eat ηi ηF ood ) Cov(γF ood , βF ood )
wF ood
+
+
+
+
+
+
+
where θ = (γij , βi , γM eat , βM eat , γF ood , βF ood ). Again, we ought to estimate the
three levels simultaneously because we need the covariances between parameters
at different stages.
H(T )
Finally, let g(Z) = ijt
ηM eat,t × H
F ood,t , i.e.,
H(T )
ij
H
= H
ijt + wjt × ηit × M eat,t + wjt × wM eat,t × ηit ×
=
(−IA + γij /wi + wj )
+
wj (1 + βi /wi )(−1 + γM eat /wM eati + wM eat )
+
wj wM eat (1 + βi /wi )(1 + βM eat /wM eat )(−1 + γF ood /wF ood + wF ood )
We can approximate the variance of the Hicksian total price elasticity by
V
H(T )
arθ ij
≈
1
wi
2
V ar(γij )
2
wj H
wj wM eat
M eat +
ηM eat H
V ar(βi )
F ood
wi
wi
2
wj
ηi V ar(γM eat )
wM eat
2
wj ηi H
V ar(βM eat )
F ood
2
wj wM eat
ηi ηF ood V ar(γF ood )
wF ood
wj H
wj wM eat
H
2
+
ηM eat F ood Cov(γij , βi )
(wi )2 M eat
(wi )2
+
+
+
+
+
Revista Colombiana de Estadística 36 (2013) 23–42
32
Andrés Ramírez
wj
ηi Cov(γij , γM eat )
2
wi wM eat
wj H
2
ηi Cov(γij , βM eat )
wi F ood
wj wM eat
2
ηi ηF ood Cov(γij , γF ood )
wi wF ood
wj H
wj wM eat
wj
H
2
ηM eat F ood
ηi Cov(βi , γM eat )
+
wi M eat
wi
wM eat
wj H
wj wM eat
2
M eat +
ηM eat H
w j η i H
F ood
F ood Cov(βi , βM eat )
wi
wi
wj H
wj wM eat
wj wM eat
M eat +
ηM eat H
η
η
2
i F ood Cov(βi , γF ood )
F ood
wi
wi
wF ood
wj
ηi wj ηi H
2
F ood Cov(γM eat , βM eat )
wM eat
wj
wj wM eat
2
ηi
ηi ηF ood Cov(γM eat , γF ood )
wM eat
wF ood
wj wM eat
2 wj ηi H
η
η
i F ood Cov(βM eat , γF ood )
F ood
wF ood
+
+
+
+
+
+
+
+
+
3. Results
The model is estimated using quarterly data for the period 1998-2007. The
time series data for prices and per-capita consumption of beef, poultry and pork
are taken from Federación Colombiana de Ganaderos (FEDEGAN). Data for percapita expenditures are obtained from the Colombian National Accounts (DANE
2007). Prices are built from the implicit price indices formed as the ratio between
nominal and real expenditures, i.e., Paasche indices.
We should use the True Cost of Living index, but Deaton & Muellbauer (1980b)
considered Taylor’s expansion of the cost function to show that a first order approximation to the True Cost of Living index will be the Paasche like index (see
equation 3). An empirical evidence that supports this argument is that most price
indices are highly correlated (Edgerton 1997).
Table (1) indicates that food expenditure is 25% of per-capita expenditure,
of which expenditure on meat is 30%, and finally beef expenditure is 60% of the
latter. Thus, beef consumption accounts for 4.5% of per-capita expenditure.
Historical data indicate that meat budget shares of the various types of meat
have not changed. Between 1998 and 2007 average quarterly consumption of beef
declined from 5.75 to 4.44 kg/capita, while poultry consumption rose from 2.92
to 5.49 kg/capita and pork consumption increased from 0.63 to 0.92 kg/capita.
It seems likely that this shift in consumption has been caused by changes in the
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Multi-Stage AIDS for Beef Demand in Colombia
Table 1: Descriptive Statistics: Colombian beef demand, 1998:I-2007:IV.
Variable
XT otalExpenditure
wF ood
pF ood
pN oF ood
XF oodExpenditure
wM eat
pM eat
pOtherF ood
wBeef
pBeef
wP ork
pP ork
wP oultry
pP oultry
∗
Mean
Standard Deviation
Upper level
880,923
228,820
0.25
0.0068
114.42
20.78
112.53
20.10
Middle level
218,812
50,975
0.30
0.02
128.28
29.87
104.01
16.28
Lower level
0.60
0.27
8,598
2,672
0.08
0.01
8,007
1,802
0.32
0.02
5,299
726
Jarque-Bera Test∗
0.27
0.09
0.35
0.31
0.33
0.09
0.36
0.33
0.67
0.15
0.13
0.29
0.71
0.15
p-value
Source: Author’s Estimations
relative prices of the different kinds of meat, as the data indicate that over the
period, the price index of beef rose by 200%, while the price index of poultry
increased by only 47% and the index of pork 110% (see Figures 1 and 2).
Unit root tests (Kwiatkowski, Phillips, Schmidt & Shin 1992, Ng & Perron
2001) were carried out, which indicate that all of the data series are I(1) (See
Table 2). In order to account for endogeneity, the Johansen (1988) cointegration
test was carried out at each budgeting allocation level based on equations (1).1 As
can be seen in Table 3, we cannot reject the null hypothesis of one cointegration
vector in each equation. On the other hand, we use Hayes, Wahl & Williams
(1990) statistical tests for testing weak separability on the second stage, i.e. meat
decision. We use a Wald test under the null hypothesis of weak separability, and
we cannot reject it, the p-value is 0.17.
We estimate simultaneously long-term system equations (1) for the three stages
through Generalized Method of Moments.2 In all stages, the Laspeyres index is
used to build moment conditions, because of endogeneity caused due to the Stone
index uses shares in its construction and it is not invariant to changes in units of
1 Information criteria was used to select VEC order and deterministic components of the
cointegration test.
2 Residuals are normal and homoscedastic, but because of autocorrelation, we estimate the
covariance matrix through consistent process (Newey & West 1987). Outcomes can be seen in
Table 4.
Revista Colombiana de Estadística 36 (2013) 23–42
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Andrés Ramírez
Poultry
Pork
15
10
0
5
Annual consumption (kg)
20
Beef
1998
2000
2002
2004
2006
2008
year
350
Figure 1: Meat per-capita annual consumption: Colombia, 1998:I-2007:IV.
Poultry
Pork
250
200
100
150
Price Index
300
Beef
1998
2000
2002
2004
2006
2008
year
Figure 2: Meat’s price index: Colombia, 1998:I-2007:IV.
measurement. We imposed homogeneity and symmetry conditions due to these
conditions being important for demand theory, and not always being treated as
verifiable conditions (Parikh 1988).
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Multi-Stage AIDS for Beef Demand in Colombia
35
Table 2: Unit root tests: Colombian beef demand, 1998:I-2007:IV.
Critical
N g − P erronb
Critical
Value (5%)
Value (5%)
Upper level
wF ood
0.625
0.463
-6.712
-8.100
∆wF ood
0.306
0.463
-14.062
-8.100
Log(X/P )
0.168
0.146
-2.835
-2.910
∆Log(X/P )
0.134
0.146
−2.116c
-2.910
Log(pF ood /PN oF ood )
0.192
0.146
-0.886
-2.910
∆Log(pF ood /PN oF ood )
0.144
0.146
-2.919
-2.910
Middle level
wM eat
0.482
0.463
-1.193
-8.100
∆wM eat
0.288
0.463
-13.205
-8.100
Log(XF ood /PF ood )
0.173
0.146
-0.363
-2.910
∆Log(XF ood /PF ood )
0.100
0.146
-3.004
-2.910
Log(pM eat /pN oM eat )
0.165
0.146
-2.619
-2.910
∆Log(pM eat /pN oM eat )
0.075
0.146
-2.956
-2.910
Lower level
wBeef
0.667
0.463
-3.629
-8.100
∆wBeef
0.096
0.463
-18.851
-8.100
wP ork
0.652
0.463
-2.552
-8.100
∆wP ork
0.114
0.463
-13.998
-8.100
Log(XM eat /PM eat )
0.185
0.146
-1.589
-2.910
∆Log(XM eat /PM eat )
0.089
0.146
−2.713c
-2.910
Log(pBeef /pP oultry )
0.660
0.463
-1.760
-2.910
∆Log(pBeef /pP oultry )
0.186
0.463
-3.079
-2.910
Log(pP ork /pP oultry )
0.830
0.463
−3.566c
-2.910
∆Log(pP ork /pP oultry )
0.400
0.463
-4.116
-2.910
Notes: a Null hypothesis stationarity. b Null hypothesis unit root.
c We use the M Z d statistic. However, the 5% critical value of the M P T d statistic is 5.480
t
while its values are equal to 10.161, 6.205 and 3.754 for ∆Log(X/P ), ∆Log(XM eat /PM eat )
and Log(pP ork /pP oultry ), respectively. Additionally, the 5% critical value of the M SB d
statistic is 0.168 while its values are equal to 0.235, 0.182 and 0.136 for ∆Log(X/P ),
∆Log(XM eat /PM eat ) and Log(pP ork /pP oultry ), respectively.
Source: Author’s Estimations
Variable
KP SS a
Long-term elasticities associated with each level are calculated using equations
(8), (9) and (10). Equations (12), (13) and (14) are used to calculate total longterm elasticities. As can be seen in Table (5), beef, pork and poultry are luxuries,
although this is not the result obtained for poultry if one only looked at within
meat group elasticity. On the other hand, meat expenditure elasticity is 2.16, but
its total expenditure elasticity is 1.65.3 Although it is less than one, the food
expenditure elasticity is still high at 0.76.
The partial beef expenditure elasticity is 1.07 in the Colombian economy (see
Table 5). This value is smaller than the elasticity found in Mexico which is 1.30
(Golan et al. 2000). In general, the wealthier countries in the West have expenditure elasticities of beef below 1.0 (Clark 2006, Barreira & Duarte 1997, MAFF
2000, Sulgham & Zapata 2006), while the poorer countries in the East have elas3 This
is calculated as 2.16 (within expenditure elasticity) × 0.76 (food expenditure elasticity).
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Andrés Ramírez
Table 3: Cointegration tests: Colombian beef demand, 1998:I-2007:IV.
Equation
Ho: CE(s)
Max. Eigenvaluea
Critical
Value (5%)
Traceb
Critical
Value (5%)
Upper Level
43.72
24.25
57.76
35.01
9.85
17.14
14.03
18.39
4.18
3.84
4.18
3.84
Middle Level
Meat Demandc
r=0*
33.87
24.25
51.82
35.01
r=1
11.98
17.14
17.94
18.39
r=2*
5.95
3.84
5.95
3.84
Lower Level
Beef Demandd
r=0*
36.24
24.15
57.09
40.17
r=1
13.08
17.79
20.84
24.27
r=2
5.36
11.22
7.75
12.32
r=3
2.39
4.12
2.39
4.12
Pork Demandd
r=0*
32.51
24.15
51.14
40.17
r=1
11.86
17.79
18.62
24.27
r=2
6.20
11.22
6.76
12.32
r=3
0.56
4.12
0.56
4.12
a Null hypothesis: the number of cointegrating vectors is r against the alternative of r + 1
b Null hypothesis: the number of cointegrating vectors is less than or equal to r against
general alternative
c There is a constant and a deterministic trend in the cointegrated equations.
Schwarz criterion supports these outcomes.
d There is not a constant nor a deterministic trend in the cointegrated equations.
Schwarz criterion supports these outcomes.
* Denotes rejection of the hypothesis at the 5% level
Source: Author’s estimations.
Food Demandc
r=0*
r=1
r=2*
ticities above 1.0 (Chern et al. 2003, Liu et al. 2008, Ma et al. 2003, Rastegari &
Hwang 2007).
Table 4: Residuals tests: Colombian beef demand, 1998:I-2007:IV.
Breusch-Pagan-Godfreyb
Upper Level
Food Demand
1.61
3.31
Middle Level
Meat Demand
2.34
8.70
Lower Level
Beef Demand
1.24
6.66
Pork Demand
2.75
5.72
a The null hypothesis is normality
b The null hypothesis is homocedasticity
c The null hypothesis is not autocorrelation
* Denotes rejection of the hypothesis at the 5% level
Source: Author’s estimations.
Equation
Jarque-Beraa
Breusch-Godfreyc
36.21*
27.82*
24.53*
30.26*
As can be seen in Table (6), there is substitution of poultry for beef within
the meat group, but this effect is not present if taking into account that a change
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Multi-Stage AIDS for Beef Demand in Colombia
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Table 5: Expenditure elasticities for the three levels: Colombian beef demand, 1998:I2007:IV.
Upper level
Other goods
1.07*
(0.011)
Middle level
Meat
Other food
2.16*
0.48*
(0.291)
(0.129)
Lower level
Within meat group
Beef
Pork
Poultry
1.07*
1.78*
0.64*
(0.145)
(0.367)
(0.268)
Total
Beef
Pork
Poultry
1.78*
2.95*
1.05*
(0.378)
(0.687)
(0.166)
Standard deviation are calculated with Delta method.
∗ Significant at 5%
Source: Author’s estimations
Food
0.76*
(0.034)
of poultry price implies reallocation effects of meat within food and food within
total consumption. With regard to total uncompensated and compensated ownprice elasticities, we can see that beef is quite elastic, and the differences between
within meat group and total elasticities are large. This fact can be misleading if
the within elasticities are used for making policy judgements.4
The partial own-Marshallian price demand elasticity is −1.17 in Colombia (see
Table 6). This value is similar to elasticities that are internationally found (Galvis
2000, Golan et al. 2000, Fousekis & Revell 2000). Usually, this elasticity is less than
−1. With regard to the partial compensated price elasticity, it is found a value
equal to −0.52 in the Colombian economy (see Table 6). The partial own-Hicksian
price demand elasticity is −0.59 in Mexico (Golan et al. 2000). This elasticity
internationally has a range between −0.23 and −1.63. The highest elasticity in
absolute value is found in Nigeria (Osho & Nazemzadeh 2005), while the lowest is
found in U.S. (Asatryan 2003).
4 Uncompensated own-price elasticities of poultry and pork are −1.020 and −0.028, respectively.
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Andrés Ramírez
Table 6: Uncompensated and compensated beef price elasticities: Colombian beef demand, 1998:I-2007:IV.
Marshallian
Hicksian
Beef
Pork
Poultry
Beef
Pork
Within
-1.17*
-0.04
0.14*
-0.52*
0.04*
(0.142)
(0.033)
(0.043)
(0.063)
(0.001)
Total
-1.95*
-0.09
-0.03
-1.78*
-0.08
(0.278)
(0.047)
(0.111)
(0.262)
(0.045)
Standard deviation are calculated with Delta method.
∗ Significant at 5%
Source: Author’s estimations
Poultry
0.47*
(0.003)
8E-03
(0.103)
Table 7: Short-term beef elasticities: Colombian beef demand, 1998:I-2007:IV.
Beef demand
Total expenditure
Beef price
Pork price
0.034
-0.247
-0.025
Source: Author’s estimations
Poultry price
0.103
4. Simulations
In order to calculate short-term elasticities, Seemingly Unrelated Regression
Equations are used for estimating an Error Correction Linear AIDS with the three
stages simultaneously. Monte Carlo simulation exercises are done based on the
estimated model in order to analyse the short-term dynamics of beef demand. The
algorithm used solves the model for each observation in the solution sample, using
a recursive procedure to compute values for the endogenous variables. The model
is solved repeatedly for different draws of the stochastic components (coefficients
and errors). During each repetition, errors are generated for each observation in
accordance with the residual uncertainty in the model. The three stages are linked
by prices and expenditures; for example, a shock on consumption expenditure
causes a direct effect on food demand, which implies an expenditure effect on
meat demand, and as consequence a reallocation within the group. On the other
hand, a change of beef price implies a direct effect within the meat group, but also
affects meat within food and food within consumption.
The simulation results suggest a good fit for each equation in the model; during
the period analysed observed data fell inside the 95% prediction interval (outcomes
upon author’s request).
We analyse transitory effects associated with a positive shock on total expenditure, and increases in beef, poultry and pork prices. We use our simulated model
to measure the impact on beef demand by comparing in-sample forecasted beef
demand with and without the shocks for the first quarter of 2007. Given that
a comparison is being performed, the same set of random residuals is applied to
both scenarios during each repetition. This is done so that the deviation between
the different scenarios is based only on differences in the exogenous variables, not
on differences in random errors.
Revista Colombiana de Estadística 36 (2013) 23–42
Multi-Stage AIDS for Beef Demand in Colombia
39
The first exercise evaluates the short-term effect on beef demand associated
with a positive shock on total expenditure. Specifically, we increase the consumer
expenditure by 1%, and compare this scenario with the baseline scenario (without
shock). We find that there is an increase in beef demand by only 0.034%. On
the other hand, we evaluate the short-term effects in beef demand associated with
transitory increases in beef, pork and poultry prices. It can be seen on Table 7,
that an increase of 1% in beef price reduces its own demand by 0.24%. Finally,
there is a substitution effect of poultry for beef, because an increases of 1% on
poultry price causes an increase in beef demand by 0.1%, while an increase in
pork price causes very little effect on beef demand.
5. Conclusions
The results in the long-term indicate that the expenditure elasticity of food is
less than one, supporting the idea of a normal good. On the other hand, meat is
a luxury good because its expenditure elasticity is greater than one. In the lower
level, the cross price elasticities indicate that there is a bigger substitution effect
of beef for poultry than beef for pork. Although the total expenditure elasticity of
demand for beef is 1.78 in the long-term, the short-term expenditure elasticity is
merely 0.034. The smaller short-term reaction of the consumers is also evidenced
in price shocks; while the own price elasticity of beef is −0.24 in the short-term,
the long-term total elasticity is −1.95. These differences between elasticities obey
the small velocities of convergence in the three levels of the model. Specifically,
the velocities of convergence are 2%, 10% and 17% on the beef, meat and food
demand equations.
Colombian real per-capita total expenditure has grown at 2.1% per annum from
2000 to 2007; therefore, given a 1.5% population growth rate per annum, the total
expenditure beef elasticity implies beef demand growing at 5.3% a year.5 However,
Colombian beef production has grown at −0.51% per annum in the same period,
this difference has caused Colombian beef price to increase by 14.7% per annum.
Recently, Colombia has been negotiating international trade agreements with the
United States and the European Union. This implies that the Colombian beef
sector would have international competition from countries with high subsidies,
and as a consequence, the internal beef price would decrease. These facts would
have important effects on domestic producers, which ought to improve productivity
in order to stay as an important sector in the Colombian economy and make good
use of the new market opportunities.
Recibido: abril de 2012 — Aceptado: abril de 2013
5 This is calculated as 1.5% (population growth rate per annum) + 2.1% (per-capita total
expenditure growth per annum) * 1.78 (total expenditure elasticity).
Revista Colombiana de Estadística 36 (2013) 23–42
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Andrés Ramírez
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